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Concept# Option style

Résumé

In finance, the style or family of an option is the class into which the option falls, usually defined by the dates on which the option may be exercised. The vast majority of options are either European or American (style) options. These options—as well as others where the payoff is calculated similarly—are referred to as "vanilla options". Options where the payoff is calculated differently are categorized as "exotic options". Exotic options can pose challenging problems in valuation and hedging.
American and European options
The key difference between American and European options relates to when the options can be exercised:

- A European option may be exercised only at the expiration date of the option, i.e. at a single pre-defined point in time.
- An American option on the other hand may be exercised at any time before the expiration date.

- \max{(S-K), 0}, for a call option
- \max{(K-S), 0},

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FIN-522: Venture capital

The course applies finance tools and concepts to the world of venture capital and financing of projects in high-growth industries. Students are introduced to all institutional aspects of the venture capital industry. Students analyze various aspects of VC finance using an investors' perspective.

FIN-472: Computational finance

Participants of this course will master computational techniques frequently used in mathematical finance applications. Emphasis will be put on the implementation and practical aspects.

FIN-404: Derivatives

The objective of this course is to provide a detailed coverage of the standard models for the valuation and hedging of derivatives products such as European options, American options, forward contracts, futures contract and exotic options.

I present a tractable framework, first developed in Trolle and Schwartz (2009), for pricing energy derivatives in the presence of unspanned stochastic volatility. Among the model features are i) a perfect fit to the initial futures term structure, ii) a fast and accurate Fourier-based pricing formula for European-style options on futures contracts, enabling efficient calibration to liquid plain-vanilla exchange-traded derivatives, and iii) the evolution of the futures curve being described in terms of a low-dimensional affine state vector, making the model ideally suited for pricing complex energy derivatives and real options by simulation. I also consider an extension of the framework that takes jumps in spot prices into account.

In the first chapter,which is a joint work with Mathieu Cambou and Philippe H.A. Charmoy, we study the distribution of the hedging errors of a European call option for the delta and variance-minimizing strategies. Considering the setting proposed by Heston (1993), we assess the error distribution by computing its moments under the real-world probability measure. It turns out that one is better off implementing either a delta hedging or a variance-minimizing strategy, depending on the strike and maturity of the option under consideration. In the second paper, which is a joint work with Damir Filipovic and Loriano Mancini, we develop a practicable continuous-time dynamic arbitrage-free model for the pricing of European contingent claims. Using the framework introduced by Carmona and Nadtochiy (2011, 2012), the stock price is modeled as a semi-martingale process and, at each time t , the marginal distribution of the European option prices is coded by an auxiliary process that starts at t and follows an exponential additive process. The jump intensity that characterizes these auxiliary processes is then set in motion by means of stochastic dynamics of Itô's type. The model is a modification of the one proposed by Carmona and Nadtochiy, as only finitely many jump sizes are assumed. This crucial assumption implies that the jump intensities are taken values in only a finitedimensional space. In this setup, explicit necessary and sufficient consistency conditions that guarantee the absence of arbitrage are provided. A practicable dynamic model verifying them is proposed and estimated, using options on the S&P 500. Finally, the hedging of variance swap contracts is considered. It is shown that under certain conditions, a variance-minimizing hedging portfolio gives lower hedging errors on average, compared to a model-free hedging strategy. In the third and last chapter, which is a joint work with Rémy Praz, we concentrate on the commodity markets and try to understand the impact of financiers on the hedging decisions. We look at the changes in the spot price, variance, production and hedging choices of both producers and financiers, when the mass of financiers in the economy increases. We develop an equilibrium model of commodity spot and futures markets in which commodity production, consumption, and speculation are endogenously determined. Financiers facilitate hedging by the commodity suppliers. The entry of new financiers thus increases the supply of the commodity and decreases the expected spot prices, to the benefits of the end-users. However, this entry may be detrimental to the producers, as they do not internalize the price reduction due to greater aggregate supply. In the presence of asymmetric information, speculation on the futures market serves as a learning device. The futures price and open interest reveal different pieces of private information regarding the supply and demand side of the spot market, respectively. When the accuracy of private information is low, the entry of new financiers makes both production and spot prices more volatile. The entry of new financiers typically increases the correlation between financial and commodity markets.

Semyon Malamud, Alberto Mokak Teguia

We characterize the unique equilibrium in an economy populated by strategic CARA investors who trade multiple risky assets with arbitrarily distributed payoffs. We use our explicit solution to study the joint behavior of illiquidity of option contracts. Option bid-ask spreads are proportional to risk aversion and risk-neutral variances of option payoffs. Spreads may decrease in risk aversion, physical variance, open interest, and increase after earnings announcements in a result contrary to conventional wisdom. All these predictions are confirmed empirically using a large panel data set of U.S. stock options.